Problem: Find the remainder when $x^4 + 1$ is divided by $x^2 - 3x + 5.$
Answer: The long division is shown below.

\[
\begin{array}{c|cc ccc}
\multicolumn{2}{r}{x^2} & +3x & +4 \\
\cline{2-6}
x^2 - 3x + 5 & x^4 & & & & +1 \\
\multicolumn{2}{r}{x^4} & -3x^3 & +5x^2 \\
\cline{2-4}
\multicolumn{2}{r}{} & +3x^3 & -5x^2 & \\
\multicolumn{2}{r}{} & +3x^3 & -9x^2 & +15x \\
\cline{3-5}
\multicolumn{2}{r}{} & & +4x^2 & -15x & +1 \\
\multicolumn{2}{r}{} &  & +4x^2 & -12x & +20 \\
\cline{4-6}
\multicolumn{2}{r}{} &  &  & -3x & -19 \\
\end{array}
\]Thus, the remainder is $\boxed{-3x - 19}.$